Questions tagged [sequence-of-function]

Use this tag only when your query is about sequences of functions. Don't use this tag for any other sequence such as sequences of real numbers or sequences of complex numbers etc.

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42 views

Similar to Dini's Theorem, without the hypothesis of the continuity of $f_{n}$? Is true?

Let $(f_{n})_{n \in \mathbb{N}}$ be a sequence of non-decreasing function of $[0,1]$ to $[0,1]$. Assume that (1) For any $x \in [0,1]$, the pointwise limit exist and $f(x) = \lim_{n \to \infty}f_{...
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1answer
179 views

Proof of Montel's theorem Stein and Shakarchi

I am reading the proof of Montel's theorem in Stein and Shakarchi Complex Analysis book. I am stuck in page 227, at the start of the proof where we show $\mathcal{F}$ is a normal family. The proof ...
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1answer
119 views

Is it true that $\lim_{n\to\infty}\int_{0}^{1}f_n(x)\,dx = \int_{0}^{1}f(x)\,dx$ in general and if $|f_n(x)|\le 2017$?

Let $f_n(x)$ and $f(x)$ be continuous functions on $[0, 1]$ such that $\lim_{n\to\infty} f_n(x) = f(x)$ for all $x \in [0, 1]$. Answer each of the following questions. If your answer is “yes”, then ...
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1answer
39 views

Continuous mapping theorem for a sequence of densities?

Let ${f_n(x)}$ be a sequence of densities that uniformly converges to $f(x)$ almost surely, that is, $$ f_n(x) \xrightarrow[]{\text{a.s.}} f(x), \quad \text{uniformly},$$ or equivalently $$ \Pr\...
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3answers
110 views

Find a polynomial function f(n) that follows the following sequence: 1,1,2,4,7,11,16,21

my question is as follows: Find a polynomial function $f(n)$ such that $f(1),f(2),…,f(8)$ is exactly the following squence: 1,1,2,4,7,11,16,22. (Hint: how does the sum $\sum_{n=2}^{i=0}i$ come into ...
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1answer
55 views

infinite sum of holomorphic functions

Let $H$ be the open upper half plane. Let $\{f_n\}$ be a sequence of holomorphic functions in $H$ and continuous in $\bar{H}$. Suppose $\lim_{|z| \rightarrow \infty} f_n(z)$ exists and $|f_n(x)| \leq ...
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2answers
85 views

For every continuous function $g$ does there exists a sequence of functions $\{f_n\}$ which converges to $g$?

Let, $C(\Bbb R):=\\ \{f:\Bbb R \to \Bbb R| \text{ $f$ is continuous and $\exists$ a compact set $K$ such that $f(x)=0$, $\forall x\in K^c$}\}.$ Let, $\displaystyle g(x)=e^{-x^2}$ for all $x\in \Bbb R$...
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1answer
63 views

Where is the mistake in my counter-example to the statement? (sequences of functions, uniform convergence)

Statement: Let $f,f_n:[a,b]\to\mathbb{R}$ for $n\in\mathbb{N}$ and for all $x\in[a,b]$ let $\lim_{n\to\infty}f_n(x) = f(x)$. If $f$ and all $f_n$ are continuous, then $f_n$ converges uniformly ...
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3answers
69 views

Derivative of the limit of function sequence

Suppose $(f_n)$ is a sequence of real functions that converges uniformly to $f:\mathbb R\to\mathbb R$. Is it possible to show that $$f'(x)=\lim_{n\to\infty}\frac{f_n(x+1/n)-f_n(x)}{1/n},$$ and in that ...
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2answers
41 views

How can I prove this sequence of functions pointwisely converges to zero using formal argument?

Let $f_n(x)$ be a sequence of functions defined on $ [0,1]$ by $$\ f_n(x)= \begin{cases} n \text { if $0< x < \frac{1}{n}$}, \\ 0 \text { if $x=0$ or $ \frac{1}{n}\le x \le 1$ }\end{...
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0answers
27 views

Proof that a sequence of numbers must always contain a perfect square [duplicate]

Given $f(n) = n + \left \lfloor{\sqrt n}\right \rfloor$ Prove that for any natural number $n$, the sequence $n, f(n), f(f(n)),.......$ must always contain at least one perfect square. Now obviously ...
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1answer
97 views

Example of cut off function

For any $R>0,$ I need an example of a family of functions $\phi_R\in C_c^{1}(\mathbb{R}^N)$ such that $0\leq \phi_R\leq 1$ in $\mathbb{R}^N$ satifying $\phi_R=1$ in $B(0,R)$ and $\phi_R=0$ in $\...
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2answers
65 views

prove that if a series of functions converges uniformly in D… [closed]

I met this question searching online for exercises but I can't seem to solve it. Prove that if the series of functions $\sum_{n=1}^\infty f_n(x)$ converges uniformly in group D, then the sequence of ...
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0answers
15 views

Equicontinuity with respect to a sequence of points? (terminology question)

I'm searching for an established term for "equicontinuity with respect to a sequence of points" for a function. Let $\{f_k\}$ be a family of functions and $\{x_k\}$ some associated points. My version ...
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1answer
21 views

Sums of equidifferentiable functions are themselves equidifferentiable?

Suppose the sequence of vector valued functions $\{ {\bf f}_n \}$ are equidifferentiable at ${\bf x}_0$. In other words: $$\lim_{{\bf h} \to {\bf 0}} \max_n \frac{\left\Vert {\bf f}_n({\bf x}_0+{...
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1answer
36 views

Functions sequence derivative and Fourier transform analysis through distributions

Let $f_n : \mathbb{R} \to \mathbb{C}$ $$\begin{align}&f_n(x) = n \sin n \pi x \quad 0 \leq x \leq 1 \\ &f_n(x) = 0 \quad x \lt 0 \lor x \gt 1\end{align}$$ Can you calculate Fourier transforms ...
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1answer
145 views

On uniform convergence of functions with compact support [closed]

If sequence of functions $f_{n}$ in $C_C(\mathbb{R})$ converges pointwise to the functions $e^{-x^2}$, then can we conclude $f_n$ converge uniformly.
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2answers
165 views

Proof that a sequence converges pointwisely, but not uniformly

I have the following task in my homework: We consider a sequence of functions $(f_n)_{n\in\mathbb{N}}$ given by $f_n: \mathbb{R} \to \mathbb{R}$ for all $n\in\mathbb{N}$. Prove that $f_n(x) = 1 - χ_{...
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1answer
61 views

Uniform convergence on two separate intervals

Test the sequence of functions for uniform convergence $$f_n(x)=\frac{nx}{1+n^2x^2}$$ $a) \text{on} \ [0,1]$ $b) \text{on} \ [1,{\infty})$ a) If we differentiate this with respect to ...
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4answers
56 views

Evaluating the following limit

Compute the following limit: $\lim\limits_{n\to \infty}2n \int \limits_0^1\dfrac{x^{n-1}}{1+x}\,dx.$ The value is 1. I have done this by squeezing it. Is there any other way to evaluate this ?
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1answer
73 views

Uniform convergence and boundedness for the sequence of functions $\{f_{n}\}$

For $n \geq 1$, let $f_{n}(x) = x e^{-nx^2}$, $x \in \Bbb{R}$; Then the sequence $\{f_{n}\}$ is Uniformly convergent on $\Bbb{R}$?. I did this $f_{n}(x) \rightarrow 0$ as $n \rightarrow \infty$ for ...
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3answers
39 views

Pointwise and Uniform Convergence on a specific intervall

I have the sequence of functions $$ f_n(x) = \frac{x}{x^2+ \frac{1}{n}} \quad x \in [0, \infty)$$ and I need to show that the sequence converges pointwise as well as uniformly, however only on the ...
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1answer
22 views

Limit of the series of functions $f_n$

$$f_n(x)= \begin{cases} 1&\text{if }\, x\geq 1/n\\ n|x|&\text{if }\, x< 1/n. \end{cases}$$ We want to find the pointwise limit of this function. I think the answer should be $f(x)=1\...
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1answer
81 views

Limit of $\sin(kx)$ as k tends to infinity

I am have been thinking lately of the sequence of functions $$ f_n = \sin nx $$ and its limit as n tends to infinity. I am quite comfortable with the fact that viewing this sequence in $\mathcal{C}([...
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1answer
183 views

Equidifferentiable iff derivative is equicontinuous?

Let $\{f_n\}$ a sequence of function differentiable at $x_0$ We have equidifferentiability at $x_0$, if $\lim_{h \to 0} \max_n \left| \frac{f_n(x_0+h) - f(x_0)}{h}- f'_n(x_0)\right| = 0$ Are the ...
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1answer
46 views

showing existence of uniform convergent subsequence of functions (Arzela-ascoli applying?)

Let $\{f_n\}$ be a sequence of real-valued $C^1$ functions on [0,1] such that for all n, $|f'_n(x)|\le 1/\sqrt x$ ($0<x\le 1$) and $\int_{0}^{1}f_n(x)dx=0$ prove that $\{f_n(x)\}$ has a ...
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1answer
51 views

Sequence of functions and improper integrals

Let $\{f_n\}$ be a sequence of $\textbf{continuous}$ real valued functions defined on $[0, \infty)$. Suppose $f_n(x) \rightarrow f(x)$ for all $x \in [0,\infty)$ and that $f$ is integrable. Then ...
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1answer
21 views

Cauchy sequence of Functions

Formally, given a metric space $(X, d)$, a $x_1, x_2, x_3, ...$ is Cauchy, if for every positive real number $ε > 0$ there is a positive integer $N$ such that for all positive integers $...
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1answer
119 views

Converging sequence of solution to a differential equation

I was working on a problem about a sequence of functions, each of which is a solution to a sequence of differential equations, that converges to a function which is supposed to be the limit of the ...
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4answers
49 views

Prove that the following sequences of functions converges to zero

Let$\ f_n (x)=n^2x(1-x)^n$ I need to prove that$\ f_n→0$ in the interval$\ [0,1]$. Let$\ f_n(x) = nx^n$ prove that$\ f_n→0$ in the interval$\ [0,1)$. For both of these sequences I tried the ...
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1answer
37 views

Sequence of functions uniformly differentiable at a point?

Let $\{f_{n}\}$ be a sequence of functions differentiable at $x_0$. Let $r_{n}(h,x_0) = f_n(x_0+h) - f_n(x_0) - f'_n(x_0)h $ We know that for any fixed $n$, $r_n(h,x_0) = o(|h|)$ as $h \to 0$ from ...
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1answer
27 views

Family of functions over compact have derivative with the same Lipschitz constant. Do the functions themselves have the same Lipschitz constant?

Suppose a family of functions $\{f_n\}$ defined over a compact, with the $q$th derivatives ${f^{q}_n}$ Lipschitz continuous with the same Lipschitz constant for all $n$. Do the lower order derivatives ...
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1answer
51 views

Taylor Expansion for Sequence of Functions

Suppose we have a sequence of functions $\{ f_n \}$: $\mathbb{R}^p \to \mathbb{R}^q$ differentiable at $x$. Let $x_n \to x$, and consider the Taylor expansion $f_n(x_n) = f_n(x) + \nabla f_n(x)(x_n-...
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1answer
66 views

Let, $g$ be a cont. func. on $[0,1]$,$g(1)=0$,$\{f_n\}$ be a seq. of func. on $[0,1]$, $f_n (x)=x^n g(x)$. Prove,$\{f_n\}$ converge unif'ly on$[0,1]$.

Firstly, I am sorry, I am unable to put my question properly on the title dut to the character limit. The problem actually looks like- Let, $g:[0,1]\to\Bbb{R}$ be a continuous function such that $g(...
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1answer
12 views

Find the limit function for the sequence of function problem

Let us consider the sequence of function $$ f_n(x)= \begin{cases} nx, x\in [0, 1/2^n] \\ 1/nx, x \in (1/2^n, 1]\\ \end{cases} $$ Find the limt function $f(x).$ My work: if $x \in [0, 1/2^n]$ then $...
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1answer
84 views

If $f_n \rightarrow f$ is a sequence of $L^p$ and $g_n \rightarrow g$ a bounded sequence of $L^{\infty}$ then $f_ng_n \rightarrow fg$ in $L^p$

I want to verify is my proof is correct. Let $p \in [1,\infty)$, $f_n$ a sequence of $L^p$ that converges to $f$ and $g_n$ a bounded sequence of $L^\infty$ that converges almost everywhere to $g$. ...
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2answers
344 views

Show that the sum of reciprocal products equals $n$

I don't even know how to proceed. Please help me with this. (Original at https://i.stack.imgur.com/DRIX8.jpg) Consider all non-empty subsets of the set $\{1, 2, \ldots, n\}$. For every such ...
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1answer
36 views

Convergence of sequence of functions.

Can someone explain to me why ${f_n}=x^{\frac{1}{2n-1}}$ defined on [-1,1] converges pointwise to $1$ if $x \in (0,1)$, to $-1$ if $x \in (-1,0)$, and to $0$ if $x=0$. This is the answer I got but I ...
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2answers
28 views

Normed spaces: Sequence of functions/convergence

$(f_n)$is a sequence of functions in $C([0,1])$ How to prove, that if $f_n$ converges to $ f \in C([0, 1]) $ with respect to $||·||_\infty$ then $f_n$ also converges to $f$ with respect to $ ||·||_1$...
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1answer
62 views

Pointwise convergence and continuity problem

I have the following function: $$f_{n}(x)=\frac{n}{1+nx}$$ and want to study its convergence on $[0,1]$. I let $x$ be fixed on $[0.1]$ and compute the limit of $f_{n}(x)$. I can rewrite it as $f_{n}(...
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2answers
43 views

find the limit function $\lim _{n \rightarrow \infty} P_n(x)$

Let $P_n$ be a sequence of polynomials such that for $n=0, 1, 2,...$ $P_0=0$ and $P_{n+1}(x)=P_n(x)+\frac{x^2-P_n^2(x)}{2}$. Assuming the fact that $\{P_n\}$ is convergent pointwise, find the limit ...
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2answers
137 views

Show that $f_n(x)= \frac{\sin^2(n^{\alpha}x)}{nx}$ is not uniformly convergent for $\alpha \geq 1$ and $x \neq 0$

Can someone help me with proving that \begin{align} f_n: \mathbb{R} \rightarrow \mathbb{R }, \hspace{0.5cm} f_n(x)= \begin{cases} \frac{\sin^2(n^{\alpha}x)}{nx} & \text{ if } x \neq 0, \\ 0 &...
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1answer
62 views

Issue with pointwise limit to the indicator function on $\mathbb{Q}$.

As discussed in the following question it's impossible to have a pointwise limit of continuous functions converge to $1_{\mathbb{Q}\cap[0,1]}$. That said, why doesn't the following sequence of ...
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1answer
38 views

Convergence of sequence of function $\frac{|x|^n}{n+|x|^n}$

When does this sequence of functions converges uniformly: $$f_n(x)=\frac{|x|^n}{n+|x|^n}$$ As I observed, this sequence converges to $0$ pointwise in $[-1,1]$ but I am unable to tackle the case for ...
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1answer
27 views

Proving a sequence of functions is not uniformly convergent

Study the uniform convergence of $f_n(x) = \frac{1}{n^2 x^2} , x \in (0,1].$ I'm having troubles manipulating $n$ and $x$ when trying to show whether a sequence is uniformly convergent or not. This ...
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1answer
39 views

Finding limit of a convergent sequence

Let $(X,\mathcal{A},\mu)$ be a measure space and let $f_n:X\to [0,+\infty]$ be a sequence of $\mathcal{A}$-measurable functions. For each $n\in \mathbb{N}$, we define an incresing sequence $(g_{n,k})$ ...
1
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1answer
52 views

Convergence of sequence of functions $\sum_{n\ge1 }ne^{-nx^2}$

How to show that the series $$\sum_{n\ge1 }ne^{-nx^2}$$ is not uniformly convergent for $x\in (0,\infty)$ whereas converges uniformly when $x\in (a,\infty),\ a>0$? What I observed is, for $x>a$...
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1answer
44 views

Understanding technique for proving uniform convergence of sequence of functions

Consider the sequence of functions $f_n(x)=\frac{x}{nx+1}, x\in(0,1).$ Does it converge uniformly? If $f_n$ are differentiable $\forall n$ and $f_n(x)$ converges pointwise to $f(x)$, this technique ...
2
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1answer
45 views

Proving convergence of sequence of functions of sequence

Consider the sequence of functions $f_k :A \rightarrow \mathbb{R}$. Suppose that $\{f_k\} \rightarrow f$ uniformly in A and $f_k$ is continuous in A $\forall k$. Prove that if $x_0 \in A$ and $\{x_k \...
0
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2answers
50 views

sequence of functions and continuity [closed]

I have a couple of questions on sequences of functions to help me understand the concept better. If you have a function f, what must be required of the function to be able to build a sequence of ...